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The function slowly grows to positive infinity as x increases and rapidly goes to negative infinity as x approaches 0 ("slowly" and "rapidly" as compared to any power law of x); the y-axis is an asymptote, as seen in .
Interactive Graph: Graph of the Natural Logarithm
Graph of the natural logarithm, $y=ln(x)$or $e^y=x$. The function slowly grows to positive infinity as x increases and rapidly goes to negative infinity as x approaches 0. What shape does the graph become?
as n approaches infinity. In other words, e is the sum of 1 plus 1/1 plus 1/(1*2) plus 1/(1*2*3), and so on.
The number e has many applications in calculus, number theory, differential equations, complex numbers, compoundinterest, and more. It also is extremely useful as a base in logarithms; so useful that the logarithm with base e has its own name (natural logarithm) and symbol. Here is the proper notation for the natural logarithm of x:
The natural logarithm is so named because unlike 10, which is given value by culture and has minimal intrinsic use, e is an extremely interesting number that often "naturally" appears, especially in calculus.
The inverse of the natural log appears, for example, upon differentiating a logarithm of any base:
Outside of calculus, the natural logarithm can be used to relate 1, e, i, and π, four of the most important numbers in mathematics:
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